(c) Suppose px increases by 50%. What is Joe’s new optimal consumption bundle? Calculateboth the Income Effect and the Substitution Effect.

Given that px increases by 50%, the new px = 2 * 1.5 = 3.

Step 1: Calculate the new optimal consumption bundle

Substitute the new px into the equations we derived for the optimal consumption bundle (x*, y*).

x* = I/(px + βpx) x = 24/(3 + 23) x = 24/9 x* = 8/3

y* = (βpx/αpy)I/(px + βpx) y* = (23/63)24/(3 + 23) y* = 12/1824/9 y = 2/38/3 y = 16/9

So, Joe's new optimal choice is to consume 8/3 units of good x and 16/9 units of good y.

Step 2: Calculate the Income Effect and the Substitution Effect

The income effect is the change in consumption due to the change in purchasing power (real income) caused by the price change. The substitution effect is the change in consumption due to the change in relative prices, holding real income constant.

To isolate the substitution effect, we need to calculate the consumption bundle at the new prices but with the purchasing power (real income) held constant at the original level. This is done by adjusting the income so that the consumer can just afford the original bundle at the new prices.

The original bundle was (4, 16/3) and the new prices are (3, 3). So, the income required to afford the original bundle at the new prices is 34 + 316/3 = 24. This is the same as the original income, so the substitution effect is the entire change in the consumption bundle, which is (8/3 - 4, 16/9 - 16/3) = (-4/3, -32/27).

The income effect is then the total change in consumption minus the substitution effect. But since the income didn't change, the income effect is (0, 0).